Belief
Dispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

Belief Dispersion and Decreasing Returns in the
Stock Market and in the Real Economy

Arrow-Debreu
equilibrium
Securities
marketsElyès Jouini

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Séminaire Bachelier, IHP, November 20, 2020

Introduction
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Shiller (1981), stock market volatility is too high relative
to the ex post variability of dividends

Introduction
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Shiller (1981), stock market volatility is too high relative
to the ex post variability of dividends
First explanation: …nancial leverage (Black, 76, Christie,82), explains only a small part

Introduction
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Shiller (1981), stock market volatility is too high relative
to the ex post variability of dividends
First explanation: …nancial leverage (Black, 76, Christie,82), explains only a small part
Heterogeneous beliefs (Basak 00, 2005, JN, 07, 11,
Bhamra-Uppal, 14, Atmaz-Basak, 18): additional sourceof risk (excessive volatility, risk premium puzzle)

Introduction
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Shiller (1981), stock market volatility is too high relative
to the ex post variability of dividends
First explanation: …nancial leverage (Black, 76, Christie,82), explains only a small part
Heterogeneous beliefs (Basak 00, 2005, JN, 07, 11,
Bhamra-Uppal, 14, Atmaz-Basak, 18): additional sourceof risk (excessive volatility, risk premium puzzle)
But exogenous dividend/production process (risk/return
trade-o¤)Introduction
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Shiller (1981), stock market volatility is too high relative
to the ex post variability of dividends
First explanation: …nancial leverage (Black, 76, Christie,82), explains only a small part
Heterogeneous beliefs (Basak 00, 2005, JN, 07, 11,
Bhamra-Uppal, 14, Atmaz-Basak, 18): additional sourceof risk (excessive volatility, risk premium puzzle)
But exogenous dividend/production process (risk/return
trade-o¤)"Although many interesting and useful results emerge
from the analysis of exchange of …xed quantities of risk
and return, an equally important set of issues arises inconnection with the fact that the …rm may vary therisk-return combination it o¤ers" Greenberg et al. (78)In this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption goodIn this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption good
Large economy : continuum of investors

In this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption good
Large economy : continuum of investors
parsimonious description of beliefs dispersion as in AB18

In this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption good
Large economy : continuum of investors
parsimonious description of beliefs dispersion as in AB18
non-vanishing beliefs dispersion : no investor dominates
the economy in relatively extreme states

In this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption good
Large economy : continuum of investors
parsimonious description of beliefs dispersion as in AB18
non-vanishing beliefs dispersion : no investor dominates
the economy in relatively extreme states

One …rm (aggregated representation of the economy)In this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption good
Large economy : continuum of investors
parsimonious description of beliefs dispersion as in AB18
non-vanishing beliefs dispersion : no investor dominates
the economy in relatively extreme states

One …rm (aggregated representation of the economy)
Approach à la Greenberg et al. (78) : the …rm chooses
among a set of risk-return combinations ("technology" forthe production of risk and return) vs neoclassical approach(capital, labor, investment,...)

In this paper
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Financial markets equilibrium model, …nite horizon,
continuous time, one consumption good
Large economy : continuum of investors
parsimonious description of beliefs dispersion as in AB18
non-vanishing beliefs dispersion : no investor dominates
the economy in relatively extreme states

One …rm (aggregated representation of the economy)
Approach à la Greenberg et al. (78) : the …rm chooses
among a set of risk-return combinations ("technology" forthe production of risk and return) vs neoclassical approach(capital, labor, investment,...)
Capture the possibility for the …rm/economy to expand
during good times and to contract during bad times.

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria
volatile risk exposure (expands/contracts with good/badnews), quadratic relationship between risk exposure andproduction level

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria
volatile risk exposure (expands/contracts with good/badnews), quadratic relationship between risk exposure andproduction level
constant returns to scale : risk adjustments couterbalance
beliefs heterogeneity impact

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria
volatile risk exposure (expands/contracts with good/badnews), quadratic relationship between risk exposure andproduction level
constant returns to scale : risk adjustments couterbalance
beliefs heterogeneity impact
decreasing returns to scale : risk premium ‡uctuations, risk
premium volatility, and excess volatility

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria
volatile risk exposure (expands/contracts with good/badnews), quadratic relationship between risk exposure andproduction level
constant returns to scale : risk adjustments couterbalance
beliefs heterogeneity impact
decreasing returns to scale : risk premium ‡uctuations, risk
premium volatility, and excess volatility

Due to risk exposure ‡uctuations

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria
volatile risk exposure (expands/contracts with good/badnews), quadratic relationship between risk exposure andproduction level
constant returns to scale : risk adjustments couterbalance
beliefs heterogeneity impact
decreasing returns to scale : risk premium ‡uctuations, risk
premium volatility, and excess volatility

Due to risk exposure ‡uctuations
skewness, excess kurtosis and momentum in the
production process

Our main results
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Existence and full characterization of Arrow-Debreu andArrow-Radner equilibria
volatile risk exposure (expands/contracts with good/badnews), quadratic relationship between risk exposure andproduction level
constant returns to scale : risk adjustments couterbalance
beliefs heterogeneity impact
decreasing returns to scale : risk premium ‡uctuations, risk
premium volatility, and excess volatility

Due to risk exposure ‡uctuations
skewness, excess kurtosis and momentum in the
production process
skewness, excess kurtosis, short-term momentum andlong-term reversal in the asset price processOther features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)Other features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)
RP & with stock prices (Campbell-Cochrane, 99)

Other features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)
RP & with stock prices (Campbell-Cochrane, 99)
Stock volatility follows a Heston-like behavior

Other features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)
RP & with stock prices (Campbell-Cochrane, 99)
Stock volatility follows a Heston-like behavior

Volatility risk premium (Carr and Wu, 09), and the RP byunit of risk explodes in bad times (Corradi et al.,13)

Other features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)
RP & with stock prices (Campbell-Cochrane, 99)
Stock volatility follows a Heston-like behavior

Volatility risk premium (Carr and Wu, 09), and the RP byunit of risk explodes in bad times (Corradi et al.,13)
Volatility & with returns at the aggregate level (e.g.,
Schwert-Stambaugh, 87) but % with returns at theindividual …rm level (Du¤ee, 95)

Other features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)
RP & with stock prices (Campbell-Cochrane, 99)
Stock volatility follows a Heston-like behavior

Volatility risk premium (Carr and Wu, 09), and the RP byunit of risk explodes in bad times (Corradi et al.,13)
Volatility & with returns at the aggregate level (e.g.,
Schwert-Stambaugh, 87) but % with returns at theindividual …rm level (Du¤ee, 95)
Testable conseq. : Sharpe and volatilities ratios bounds
and relation between …n. volatility, RP, macro volatility,
risk aversion and instantaneous average beliefOther features
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Relative disagr. highest during recessions (Bloom, 14)
RP & with stock prices (Campbell-Cochrane, 99)
Stock volatility follows a Heston-like behavior

Volatility risk premium (Carr and Wu, 09), and the RP byunit of risk explodes in bad times (Corradi et al.,13)
Volatility & with returns at the aggregate level (e.g.,
Schwert-Stambaugh, 87) but % with returns at theindividual …rm level (Du¤ee, 95)
Testable conseq. : Sharpe and volatilities ratios bounds
and relation between …n. volatility, RP, macro volatility,
risk aversion and instantaneous average belief
Continuous time and log utility : CIR++ interest rate
model

The …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

T = [0, T ] , Brownian motion (Wt )t 2T

The …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

T = [0, T ] , Brownian motion (Wt )t 2T

One consumption good produced by the …rm, delivered atdate T , and consumption at T

The …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

T = [0, T ] , Brownian motion (Wt )t 2T

One consumption good produced by the …rm, delivered atdate T , and consumption at T

Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Controled production processdytθ = m (θ t ) ytθ dt + θ t ytθ dWt , y0θ = 1

The …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

T = [0, T ] , Brownian motion (Wt )t 2T

One consumption good produced by the …rm, delivered atdate T , and consumption at T

Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Controled production processdytθ = m (θ t ) ytθ dt + θ t ytθ dWt , y0θ = 1
m (θ ) = a + bθ cθ 2 models the uncertainty/expected
growth rate trade-o¤ or how risk is transformed into return

The …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

T = [0, T ] , Brownian motion (Wt )t 2T

One consumption good produced by the …rm, delivered atdate T , and consumption at T

Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Controled production processdytθ = m (θ t ) ytθ dt + θ t ytθ dWt , y0θ = 1
m (θ ) = a + bθ cθ 2 models the uncertainty/expected
growth rate trade-o¤ or how risk is transformed into return
A change in units of measure permits to take a = 0

The …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

T = [0, T ] , Brownian motion (Wt )t 2T

One consumption good produced by the …rm, delivered atdate T , and consumption at T

Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Controled production processdytθ = m (θ t ) ytθ dt + θ t ytθ dWt , y0θ = 1
m (θ ) = a + bθ cθ 2 models the uncertainty/expected
growth rate trade-o¤ or how risk is transformed into return
A change in units of measure permits to take a = 0
Y = y :yyTθ for some θThe …rm
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

T = [0, T ] , Brownian motion (Wt )t 2T

One consumption good produced by the …rm, delivered atdate T , and consumption at T

Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Controled production processdytθ = m (θ t ) ytθ dt + θ t ytθ dWt , y0θ = 1
m (θ ) = a + bθ cθ 2 models the uncertainty/expected
growth rate trade-o¤ or how risk is transformed into return
A change in units of measure permits to take a = 0
Y = y :yyTθ for some θAD-Prices (SPD) : r.v. p and value of y given by E [py ]Consumers/Shareholder
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

There is a continuum of consumers who own the …rm

Consumers/Shareholder
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction

There is a continuum of consumers who own the …rm
Disagree about m : a δ-type shareholder believes that m isgiven by mδ (θ ) = m (θ ) + δθ and

The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

dyt = (m (θ t ) + δθ t ) yt dt + θ t yt dWtδConsumers/Shareholder
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction

There is a continuum of consumers who own the …rm
Disagree about m : a δ-type shareholder believes that m isgiven by mδ (θ ) = m (θ ) + δθ and

The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

dyt = (m (θ t ) + δθ t ) yt dt + θ t yt dWtδA δ-type shareholder has a subjective P δ with
dP δ
1 2
δ
dP = MT = exp( 2 δ T + δWT )Consumers/Shareholder
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction

There is a continuum of consumers who own the …rm
Disagree about m : a δ-type shareholder believes that m isgiven by mδ (θ ) = m (θ ) + δθ and

The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

dyt = (m (θ t ) + δθ t ) yt dt + θ t yt dWtδA δ-type shareholder has a subjective P δ with
dP δ
1 2
δ
dP = MT = exp( 2 δ T + δWT )δ takes all possible real values

Consumers/Shareholder
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction

There is a continuum of consumers who own the …rm
Disagree about m : a δ-type shareholder believes that m isgiven by mδ (θ ) = m (θ ) + δθ and

The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

dyt = (m (θ t ) + δθ t ) yt dt + θ t yt dWtδA δ-type shareholder has a subjective P δ with
dP δ
1 2
δ
dP = MT = exp( 2 δ T + δWT )δ takes all possible real values
1 γ

Consumption at T , U δ (c ) = E MTδ u (c ) , u (x ) = x1 γ

Consumers/Shareholder
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction

There is a continuum of consumers who own the …rm
Disagree about m : a δ-type shareholder believes that m isgiven by mδ (θ ) = m (θ ) + δθ and

The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

dyt = (m (θ t ) + δθ t ) yt dt + θ t yt dWtδA δ-type shareholder has a subjective P δ with
dP δ
1 2
δ
dP = MT = exp( 2 δ T + δWT )δ takes all possible real values
1 γ

Consumption at T , U δ (c ) = E MTδ u (c ) , u (x ) = x1 γ
Initial endowment of agent δ : νδN δ0 , ω 2

De…nition, existence and uniqueness
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium

De…nitionȳ , (c̄δ )δ2R , p̄ is an ADPE if 1. ȳ = arg maxY E [p̄y ] ,2. c̄Rδ = argmax U δ (c ), E [p̄c ] νδ E [p̄ ȳ ] for all δ, and3. c̄δ d δ = ȳ .
Theorem
There exists a unique ADPE given by

Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionp̄ = ȳ

γ

exp
1

c̄δ =

(k

WT ) 2, θ̄ T2v 2
1vλδ γ M δ γ p̄
p
2πγ

τ =

1
γ

, λδ = exp

WT τ bτ(2c + γ) v 2
T δ2

v2
2

k + bv 2,
τ (2c + 1)
!

+ kδ

2c+1
v 2 & with ω 2 from ∞ (for ω = 0) to 2c+γ T (for ω = ∞).

Sketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionγ

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄
1
1
1 R
ȳ = p̄ γ
M δ γ λ̄δ γ d δ

1
γ

and

Sketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

γ

1

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄ γ and
11
1 Rȳ = p̄ γ
M δ γ λ̄δ γ d δ
γ
1
1
R
γ
1
γ
δ
θ
γ
M
θ̄ = arg maxθ E yT ȳλ̄δ d δ

Sketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

γ

1

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄ γ and
11
1 Rȳ = p̄ γ
M δ γ λ̄δ γ d δ
γ
1
1
R
γ
1
γ
δ
θ
γ
M
θ̄ = arg maxθ E yT ȳλ̄δ d δWe assume ln λ̄δ quadratic in δ, i.e. λ̄δ proportional to( v 2 T ) δ2+ kδ
λδ = exp2

Sketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

γ

1

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄ γ and
11
1 Rȳ = p̄ γ
M δ γ λ̄δ γ d δ
γ
1
1
R
γ
1
γ
δ
θ
γ
M
θ̄ = arg maxθ E yT ȳλ̄δ d δWe assume ln λ̄δ quadratic in δ, i.e. λ̄δ proportional to( v 2 T ) δ2+ kδ
λδ = exp2ztθ = ytθ ȳ γ , θ̄ = arg maxθ E qT zTθ where
γ
1
1
2
R
= exp 21 (k vW2 T )qT =M δ γ λ̄δ γ d δSketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

γ

1

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄ γ and
11
1 Rȳ = p̄ γ
M δ γ λ̄δ γ d δ
γ
1
1
R
γ
1
γ
δ
θ
γ
M
θ̄ = arg maxθ E yT ȳλ̄δ d δWe assume ln λ̄δ quadratic in δ, i.e. λ̄δ proportional to( v 2 T ) δ2+ kδ
λδ = exp2ztθ = ytθ ȳ γ , θ̄ = arg maxθ E qT zTθ where
γ
1
1
2
R
= exp 21 (k vW2 T )qT =M δ γ λ̄δ γ d δWe posit a value function of the form qt zt F (t, w ),qt = Et [qT ]Sketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

γ

1

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄ γ and
11
1 Rȳ = p̄ γ
M δ γ λ̄δ γ d δ
γ
1
1
R
γ
1
γ
δ
θ
γ
M
θ̄ = arg maxθ E yT ȳλ̄δ d δWe assume ln λ̄δ quadratic in δ, i.e. λ̄δ proportional to( v 2 T ) δ2+ kδ
λδ = exp2ztθ = ytθ ȳ γ , θ̄ = arg maxθ E qT zTθ where
γ
1
1
2
R
= exp 21 (k vW2 T )qT =M δ γ λ̄δ γ d δWe posit a value function of the form qt zt F (t, w ),qt = Et [qT ]0 = Ft + µtθ̄ F + 12 Fww + σqt σtθ̄ F + σtθ̄ Fw + σqt FwSketch of the resolution
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

γ

1

1

FOC : M δ c̄δ = λ̄δ p̄ =) c̄δ = M δ γ λ̄δ p̄ γ and
11
1 Rȳ = p̄ γ
M δ γ λ̄δ γ d δ
γ
1
1
R
γ
1
γ
δ
θ
γ
M
θ̄ = arg maxθ E yT ȳλ̄δ d δWe assume ln λ̄δ quadratic in δ, i.e. λ̄δ proportional to( v 2 T ) δ2+ kδ
λδ = exp2ztθ = ytθ ȳ γ , θ̄ = arg maxθ E qT zTθ where
γ
1
1
2
R
= exp 21 (k vW2 T )qT =M δ γ λ̄δ γ d δWe posit a value function of the form qt zt F (t, w ),qt = Et [qT ]0 = Ft + µtθ̄ F + 12 Fww + σqt σtθ̄ F + σtθ̄ Fw + σqt Fw
0 = ddθ µtθ F + σqt σtθ F + σtθ Fw θ̄ = 0, 1 = F (T , w )Sketch of the resolution (continued)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

F (t, w ) of the form exp(A(t )w 2 + B (t )w + C (t )) andθ̄ (t, w ) = D (t )w + E (t )Sketch of the resolution (continued)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

F (t, w ) of the form exp(A(t )w 2 + B (t )w + C (t )) andθ̄ (t, w ) = D (t )w + E (t )(A, B, C ) satisfy a step system of ODEs, Ricatti in A thenlinear in B then C ! θ̄

Sketch of the resolution (continued)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

F (t, w ) of the form exp(A(t )w 2 + B (t )w + C (t )) andθ̄ (t, w ) = D (t )w + E (t )(A, B, C ) satisfy a step system of ODEs, Ricatti in A thenlinear in B then C ! θ̄"! #E [p̄ c̄δ ] = ϑ exp

1 δ2 v 2
2 γ 1

E exp

should be proportional to νδ

1
2

WT

k

v 2 (1

v2
1 γδ
1
1
γ

)

2

zT

Sketch of the resolution (continued)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

F (t, w ) of the form exp(A(t )w 2 + B (t )w + C (t )) andθ̄ (t, w ) = D (t )w + E (t )(A, B, C ) satisfy a step system of ODEs, Ricatti in A thenlinear in B then C ! θ̄"! #E [p̄ c̄δ ] = ϑ exp

1 δ2 v 2
2 γ 1

E exp

1
2

WT

v 2 (1should be proportional to νδb(1 γ)+δ (2c+1 )0
Gives k =
T
2c+γ
2
2
(
2c
+
γ
)
v
+(
2c
+
1
)
T
v
(
)
1
=
2
2
ω
2T c(1 γ)+γ(2c+γ)vkv 2 δ0 and

v2
1 γδ
1
1
γ

)

2

zT

Sketch of the resolution (continued)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions

F (t, w ) of the form exp(A(t )w 2 + B (t )w + C (t )) andθ̄ (t, w ) = D (t )w + E (t )(A, B, C ) satisfy a step system of ODEs, Ricatti in A thenlinear in B then C ! θ̄"! #E [p̄ c̄δ ] = ϑ exp

1 δ2 v 2
2 γ 1

E exp

1
2

WT

k

v 2 (1

v2
1 γδ
1
1
γ

should be proportional to νδb(1 γ)+δ (2c+1 )v 2 δ0 and

Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

0
Gives k =
T
2c+γ
2
2
(
2c
+
γ
)
v
+(
2c
+
1
)
T
v
(
)
1
=
2
2
ω
2T c(1 γ)+γ(2c+γ)v

Conclusion

Uniqueness : adapt Dana (1995), BDJ (2020)

)

2

zT

Homogeneous/Exogenous benchmarks
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Without divergence of opinion (ω = 0, v = ∞),
b+ δ0
θh
2c+γ

Homogeneous/Exogenous benchmarks
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Without divergence of opinion (ω = 0, v = ∞),
b+ δ0
θh
2c+γ
In general, θ̄ 0 = θ h but θ̄ t is stochastic: θ̄ t % with Wt ,and its sensitivity % with ω 2 ; θ̄ t % with δ0

Homogeneous/Exogenous benchmarks
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Without divergence of opinion (ω = 0, v = ∞),
b+ δ0
θh
2c+γ
In general, θ̄ 0 = θ h but θ̄ t is stochastic: θ̄ t % with Wt ,and its sensitivity % with ω 2 ; θ̄ t % with δ0

Exogenous benchmark : let µ and σ be given and let us
take b = 2σc, a = µ cσ2 and c ! ∞,

Homogeneous/Exogenous benchmarks
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Without divergence of opinion (ω = 0, v = ∞),
b+ δ0
θh
2c+γ
In general, θ̄ 0 = θ h but θ̄ t is stochastic: θ̄ t % with Wt ,and its sensitivity % with ω 2 ; θ̄ t % with δ0

Exogenous benchmark : let µ and σ be given and let us
take b = 2σc, a = µ cσ2 and c ! ∞,we have θ̄ (t, Wt ) = σ and dyt = µyt dt + σyt dWt(Atmaz-Basak, 2018)

Homogeneous/Exogenous benchmarks
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Without divergence of opinion (ω = 0, v = ∞),
b+ δ0
θh
2c+γ
In general, θ̄ 0 = θ h but θ̄ t is stochastic: θ̄ t % with Wt ,and its sensitivity % with ω 2 ; θ̄ t % with δ0

Exogenous benchmark : let µ and σ be given and let us
take b = 2σc, a = µ cσ2 and c ! ∞,we have θ̄ (t, Wt ) = σ and dyt = µyt dt + σyt dWt(Atmaz-Basak, 2018)
natural, since m (σ) = µ for all c and limc!∞ m (θ ) =for θ 6= σ∞The production process
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

At equilibrium, ȳt = Y (t, Wt ) where1
Y (t, w ) = Kt exp
2

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ϕ(t )θ̄ (t, w )ϕ(t ) = (2c + γ)v 2

Arrow-Debreu
equilibrium
Securities
markets2

and Kt=ϕ (t )ϕ (0)(2c + 1)(T1
2 (2c+1 )kt)

bv 2 + T b
ϕ (0)

2

!

The production process
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

At equilibrium, ȳt = Y (t, Wt ) where1
Y (t, w ) = Kt exp
2

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ϕ(t )θ̄ (t, w )ϕ(t ) = (2c + γ)v 2

Arrow-Debreu
equilibrium
Securities
markets2

and Kt=Non Markov

ϕ (t )ϕ (0)(2c + 1)(T1
2 (2c+1 )kt)

bv 2 + T b
ϕ (0)

2

!

The production process
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

At equilibrium, ȳt = Y (t, Wt ) where1
Y (t, w ) = Kt exp
2

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ϕ(t )θ̄ (t, w )ϕ(t ) = (2c + γ)v 2

Arrow-Debreu
equilibrium
Securities
markets2

and Kt=ϕ (t )ϕ (0)(2c + 1)(Tkbv 2 + T b
ϕ (0)

t)

1
2 (2c+1 )Non Markov
ln ȳs +t /ȳs : skewness > 0 and excess kurtosis for small t

2!

The production process
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

At equilibrium, ȳt = Y (t, Wt ) where1
Y (t, w ) = Kt exp
2

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ϕ(t )θ̄ (t, w )ϕ(t ) = (2c + γ)v 2

Arrow-Debreu
equilibrium
Securities
markets2

and Kt=ϕ (t )ϕ (0)(2c + 1)(Tkbv 2 + T b
ϕ (0)

t)

1
2 (2c+1 )Non Markov
ln ȳs +t /ȳs : skewness > 0 and excess kurtosis for small t
Momentum when δ0 = 0 (no bias) and γ > 1 and noreversal (Mao and Wei, 2014)

2!

The production process
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

At equilibrium, ȳt = Y (t, Wt ) where1
Y (t, w ) = Kt exp
2

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ϕ(t )θ̄ (t, w )ϕ(t ) = (2c + γ)v 2

Arrow-Debreu
equilibrium
Securities
markets2

and Kt=ϕ (t )ϕ (0)(2c + 1)(Tkbv 2 + T b
ϕ (0)

t)

1
2 (2c+1 )Non Markov
ln ȳs +t /ȳs : skewness > 0 and excess kurtosis for small t
Momentum when δ0 = 0 (no bias) and γ > 1 and noreversal (Mao and Wei, 2014)
No such e¤ects in the homogeneous/exogenous settings

2!

Long-lived securities
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model

Two long-lived securities, risky stock and riskless bond,dSt = µt St dt + σt St dWt and r = 0
Self-…nancing strategy α, dVtα = αt (µt St dt + σt St dWt )TheoremAt the ADPE

Arrow-Debreu
equilibrium
Securities
marketsµt = v 2 (2c + γ)

Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

σt =bv 2 γ + 2ck

2cWt( ϕ (t ) + Tt)2θ̄ t ,(2c + γ) v 2
θ̄ t .
( ϕ (t ) + T t )δFor each δ, 9ᾱδ s.t. V ᾱ = c̄δ , andRᾱδ d δ = 1.

As in Du¢ e and Huang (1985), the ADPE might be
implemented in a Radner equilibrium with two securities.

Drift/RP as a function of volatility(Schwert-Stambaugh 87, Campbell-Hentschel 92)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionµ&σDrift/RP as a function of volatility(Schwert-Stambaugh 87, Campbell-Hentschel 92)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionµ&σ

σ % with θ and with ωDrift/RP as a function of production level(countercyclical, Campbell-Cochrane, 99)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Evolution of average belief/dispersion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

ConclusionAverage belief as a function of production level(procyclical)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Drift, volatilities and disagreement
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

max (1,γ)+2c

σt /θ̄ t % with ω from 1 (ω = 0) to max(1,γ)+2ct /T(ω = ∞)

Drift, volatilities and disagreement
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

max (1,γ)+2c

σt /θ̄ t % with ω from 1 (ω = 0) to max(1,γ)+2ct /T(ω = ∞)µt = (γ 1)σt θ̄ t + σ2t σt δt , positively related to σtwhen controling for θ̄ t (for γ > 1 and δt = 0) Du¤ee,
1995, (contrast with no control)Drift, volatilities and disagreement
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

max (1,γ)+2c

σt /θ̄ t % with ω from 1 (ω = 0) to max(1,γ)+2ct /T(ω = ∞)µt = (γ 1)σt θ̄ t + σ2t σt δt , positively related to σtwhen controling for θ̄ t (for γ > 1 and δt = 0) Du¤ee,
1995, (contrast with no control)
σT
θ̄ T

τ

takes all possible values in

τ

2

41,

2+

q

τω 2T

τ

γ

2

+ 4τω 2T2
τ + τω T

2

when b and c take all admissible values.

τ

γ

3
5

Volatilities ratio bounds as a function of risk
aversion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Volatilities ratio bounds as a function ofdisagreement
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Volatility risk premium
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionpp
Vt = σ2t ; dVt = Dt0 + bDt1 Vt + Dt2 Vt dt + Dt Vt dW

Volatility risk premium
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionpp
Vt = σ2t ; dVt = Dt0 + bDt1 Vt + Dt2 Vt dt + Dt Vt dW
Stochastic volatility, volexogenous setting)vol = Dt (in contrast with theVolatility risk premium
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionpp
Vt = σ2t ; dVt = Dt0 + bDt1 Vt + Dt2 Vt dt + Dt Vt dW
Stochastic volatility, volexogenous setting)vol = Dt (in contrast with the

For b = 0, Heston model with perfect correlation between
the 2 sources of risk

Volatility risk premium
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionpp
Vt = σ2t ; dVt = Dt0 + bDt1 Vt + Dt2 Vt dt + Dt Vt dW
Stochastic volatility, volexogenous setting)vol = Dt (in contrast with the

For b = 0, Heston model with perfect correlation between
the 2 sources of risk
The volatility risk premium by unit of riskµ
Λt = σtt pDVt ! ∞ during recessionst(Wt = b (Tt) + kbv 2 , θ̄ t = 0, St minimal)

Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE

Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE
Log-returns exhibit positive skewness and excess kurtosis

Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE
Log-returns exhibit positive skewness and excess kurtosis
C (S, s, t ) = limh !0 h12 cov(ln SSs +s h , ln SSt +t h )Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE
Log-returns exhibit positive skewness and excess kurtosis
C (S, s, t ) = limh !0 h12 cov(ln SSs +s h , ln SSt +t h )If δ0 = 0, γ > 1 and ω or T large enough, C (S, 0, t ) > 0
for t small : short-term momentum

Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE
Log-returns exhibit positive skewness and excess kurtosis
C (S, s, t ) = limh !0 h12 cov(ln SSs +s h , ln SSt +t h )If δ0 = 0, γ > 1 and ω or T large enough, C (S, 0, t ) > 0
for t small : short-term momentum
For T or ω large enough and ξ 2 (0, 1), C (S, 0, ξT ) < 0: long-run reversal

Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE
Log-returns exhibit positive skewness and excess kurtosis
C (S, s, t ) = limh !0 h12 cov(ln SSs +s h , ln SSt +t h )If δ0 = 0, γ > 1 and ω or T large enough, C (S, 0, t ) > 0
for t small : short-term momentum
For T or ω large enough and ξ 2 (0, 1), C (S, 0, ξT ) < 0: long-run reversal
In the exogenous setting, short-term and long-run negative
autocorrelations of returns

Equilibrium stock price
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

St = G (t, ȳt ) where G (t, y ) = H (t )y K (t ) with K (t ) = σθ̄ ttand H satis…es a …rst-order ODE
Log-returns exhibit positive skewness and excess kurtosis
C (S, s, t ) = limh !0 h12 cov(ln SSs +s h , ln SSt +t h )If δ0 = 0, γ > 1 and ω or T large enough, C (S, 0, t ) > 0
for t small : short-term momentum
For T or ω large enough and ξ 2 (0, 1), C (S, 0, ξT ) < 0: long-run reversal
In the exogenous setting, short-term and long-run negative
autocorrelations of returns
In the homogeneous framework, no momentum nor
long-run reversal

Autocorrelation between date 0 and date t returns
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Leverage e¤ect and real options
The …rm as a real option
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Debt : the value of the …rm at T is given by [ST∆]+Leverage e¤ect and real options
The …rm as a real option
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction

Debt : the value of the …rm at T is given by [ST∆]+Call option price with maturity T and strike κ

The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Ctκ,T = St N (d1 ) + N d10
κ N (d2 ) + N d20
U P
P V
U p
p V
d1 = p , d10 = p , d2 = p , d20 = p ,qqQQwhere U < V are the solutions of L2 x 2 + L1 x = ln κ.

Leverage e¤ect and real options
The risk premium increases with volatility at the idividual level (Du¤ee, 95)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Leverage e¤ect and real options
Volatility smirk (Rubinstein 94, AïtSahalia-Lo 98, Foresi-Wu 05)
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Large horizon
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Taking the limit when T ! ∞, we haveθ̄ t =σtb + δt µt
γ 1
,
= b,
= 2c + 1, δ∞ = b
2c + γ σt2c + 1θ̄ tLarge horizon
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

Taking the limit when T ! ∞, we haveσtb + δt µt
γ 1
,
= b,
= 2c + 1, δ∞ = b
2c + γ σt2c + 1θ̄ tq
σ[0,t ] = 1t Var [ln St ], µ[0,t ] = 1t E[ln St ] + 21 σ2[0,t ] andθ̄ t =µ

Sharpe[0,t ] = σ[0,t ][0,t ]

Large horizon
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets

Taking the limit when T ! ∞, we haveσtb + δt µt
γ 1
,
= b,
= 2c + 1, δ∞ = b
2c + γ σt2c + 1θ̄ tq
σ[0,t ] = 1t Var [ln St ], µ[0,t ] = 1t E[ln St ] + 21 σ2[0,t ] andθ̄ t =µ

Sharpe[0,t ] = σ[0,t ][0,t ]µExtensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionσ̄t =E[σt ] , µ̄t =E[µt ] and Sharpet = σtt = b
independent of t

Large horizon
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets

Taking the limit when T ! ∞, we haveσtb + δt µt
γ 1
,
= b,
= 2c + 1, δ∞ = b
2c + γ σt2c + 1θ̄ tq
σ[0,t ] = 1t Var [ln St ], µ[0,t ] = 1t E[ln St ] + 21 σ2[0,t ] andθ̄ t =µ

Sharpe[0,t ] = σ[0,t ][0,t ]µExtensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionσ̄t =E[σt ] , µ̄t =E[µt ] and Sharpet = σtt = b
independent of t
σ[0,t ] % with t and always above σ̄t ,Large horizon
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets

Taking the limit when T ! ∞, we haveσtb + δt µt
γ 1
,
= b,
= 2c + 1, δ∞ = b
2c + γ σt2c + 1θ̄ tq
σ[0,t ] = 1t Var [ln St ], µ[0,t ] = 1t E[ln St ] + 21 σ2[0,t ] andθ̄ t =µ

Sharpe[0,t ] = σ[0,t ][0,t ]µExtensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionσ̄t =E[σt ] , µ̄t =E[µt ] and Sharpet = σtt = b
independent of t
σ[0,t ] % with t and always above σ̄t ,µ[0,t ] % with t and always below µ̄tLarge horizon
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets

Taking the limit when T ! ∞, we haveσtb + δt µt
γ 1
,
= b,
= 2c + 1, δ∞ = b
2c + γ σt2c + 1θ̄ tq
σ[0,t ] = 1t Var [ln St ], µ[0,t ] = 1t E[ln St ] + 21 σ2[0,t ] andθ̄ t =µ

Sharpe[0,t ] = σ[0,t ][0,t ]µExtensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusionσ̄t =E[σt ] , µ̄t =E[µt ] and Sharpet = σtt = b
independent of t
σ[0,t ] % with t and always above σ̄t ,µ[0,t ] % with t and always below µ̄t

Sharpe[0,t ] &% and always lower than Sharpet

Continuous time consumption/production
Short rate and yield curve
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

2

b+ δt
t
rt = ρ + c 2cprocyclical and SRt = b2c2cδ+1+1contracyclical (for c > 0)

Continuous time consumption/production
Short rate and yield curve
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

2

b+ δt
t
rt = ρ + c 2cprocyclical and SRt = b2c2cδ+1+1contracyclical (for c > 0)

Introduction

rt = ρ + xt with

The model

dxt = κ t ϑtArrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ω2p

xt + b 2c+c1

κ t = 2 t ω2 +1 and ϑt = 21 cp

xt dt + σrtω2(t ω 2 +1 )(2c+1 )2p

xt dWt ,Continuous time consumption/production
Short rate and yield curve
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

2

b+ δt
t
rt = ρ + c 2cprocyclical and SRt = b2c2cδ+1+1contracyclical (for c > 0)

Introduction

rt = ρ + xt with

The model

dxt = κ t ϑtArrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ω2p

xt + b 2c+c1

κ t = 2 t ω2 +1 and ϑt = 21 cp

xt dt + σrtpxt dWt ,ω2(t ω 2 +1 )(2c+1 )2For b = 0, shifted CIR model (CIR++, Brigo-Mercurio,06) with time-dependent parameters

Continuous time consumption/production
Short rate and yield curve
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

2

b+ δt
t
rt = ρ + c 2cprocyclical and SRt = b2c2cδ+1+1contracyclical (for c > 0)

Introduction

rt = ρ + xt with

The model

dxt = κ t ϑtArrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ω2p

xt + b 2c+c1

κ t = 2 t ω2 +1 and ϑt = 21 cp

xt dt + σrtpxt dWt ,ω2(t ω 2 +1 )(2c+1 )2For b = 0, shifted CIR model (CIR++, Brigo-Mercurio,06) with time-dependent parameters
For ρ = 0, CIR model with time-dependent parameters

Continuous time consumption/production
Short rate and yield curve
BeliefDispersion
andDecreasing
ReturnsElyès Jouini

2

b+ δt
t
rt = ρ + c 2cprocyclical and SRt = b2c2cδ+1+1contracyclical (for c > 0)

Introduction

rt = ρ + xt with

The model

dxt = κ t ϑtArrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

ω2p

xt + b 2c+c1

κ t = 2 t ω2 +1 and ϑt = 21 cp

xt dt + σrtpxt dWt ,ω2(t ω 2 +1 )(2c+1 )2For b = 0, shifted CIR model (CIR++, Brigo-Mercurio,06) with time-dependent parameters
For ρ = 0, CIR model with time-dependent parameters
Generate a rich class of shapes for the yield curve (butalways increasing in the short-run)

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature
The …rm maximizes its market value, its decisions are
impacted by …nancial markets, and …nancial markets are
impacted by its decisions in return

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature
The …rm maximizes its market value, its decisions are
impacted by …nancial markets, and …nancial markets are
impacted by its decisions in return
We retrieved some empirical regularities and derived
testable consequences

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature
The …rm maximizes its market value, its decisions are
impacted by …nancial markets, and …nancial markets are
impacted by its decisions in return
We retrieved some empirical regularities and derived
testable consequences
It may be of interest to explore

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature
The …rm maximizes its market value, its decisions are
impacted by …nancial markets, and …nancial markets are
impacted by its decisions in return
We retrieved some empirical regularities and derived
testable consequences
It may be of interest to explore
from the …nancial point of view, debt as a strategic variable

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature
The …rm maximizes its market value, its decisions are
impacted by …nancial markets, and …nancial markets are
impacted by its decisions in return
We retrieved some empirical regularities and derived
testable consequences
It may be of interest to explore
from the …nancial point of view, debt as a strategic variablefrom the corporate governance point of view, delegation of…rm’s decisions and its impact

Conclusion
BeliefDispersion
andDecreasing
ReturnsElyès Jouini
Introduction
The model
Arrow-Debreu
equilibrium
Securities
markets
Extensions
Leverage e¤ect
and real options
Large horizon
Continuous time
consumption/production

Conclusion

We have attempted to link classical …nancial asset pricing
models to the pricing, output, and investment models in
the economics literature
The …rm maximizes its market value, its decisions are
impacted by …nancial markets, and …nancial markets are
impacted by its decisions in return
We retrieved some empirical regularities and derived
testable consequences
It may be of interest to explore
from the …nancial point of view, debt as a strategic variablefrom the corporate governance point of view, delegation of…rm’s decisions and its impact
from a macroeconomic point of view, impact of real or
perceived technological changes (e.g., dot-com bubble,
data economy, Uberization, etc.) on volatilities